Becoming a dragonborn doesn't change your type and you still explicitly count as your original race. A dragonborn warforged is a construct (living construct, dragonblooded), not a truly flesh-and-blood creature. See the Mechanics of Rebirth sidebar in Races of the Dragon. A warforged would retain their ability mods, and the traits of the living construct subtype (immunities, healing penalty, etc), but would lose all other features including their slam attack, composite plating (though you still retain the Body feat if you took one), light fortification, and so on.
Warforged components work exactly like normal items, magic or otherwise. The only differences are that only warforged can use them and they're harder to equip and remove.